Ordering is Preserved on Integers by Addition

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Theorem

The usual ordering on the integers is preserved by the operation of addition:

$\forall a, b, c, d, \in \Z: a \le b, c \le d \implies a + c \le b + d$


Proof

Recall that Integers form Ordered Integral Domain.

Then from Relation Induced by Strict Positivity Property is Compatible with Addition:

$\forall x, y, z \in \Z: x \le y \implies \paren {x + z} \le \paren {y + z}$
$\forall x, y, z \in \Z: x \le y \implies \paren {z + x} \le \paren {z + y}$


So:

\(\displaystyle a\) \(\le\) \(\displaystyle b\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle a + c\) \(\le\) \(\displaystyle b + c\) Relation Induced by Strict Positivity Property is Compatible with Addition


\(\displaystyle c\) \(\le\) \(\displaystyle d\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle b + c\) \(\le\) \(\displaystyle b + d\) Relation Induced by Strict Positivity Property is Compatible with Addition


Finally:

\(\displaystyle a + c\) \(\le\) \(\displaystyle b + c\)
\(\displaystyle b + c\) \(\le\) \(\displaystyle b + d\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle a + c\) \(\le\) \(\displaystyle b + d\) Definition of Ordering

$\blacksquare$


Sources